This new set is due Monday, March 11, at the beginning of lecture.

Write a program that, given a square matrix , computes approximations to its eigenvalues using the QR-algorithm. Ideally, the user can decide the dimensions of the matrix and, more importantly, the error within which the approximations will be found. Apply your method to a matrix, and check the number of iterations the process requires.

Please turn in: The code (best if you email it to me), a write up explaining what your code does, the matrix you applied the method to, and the result. For this, you can work in groups of two or three. In case you cannot find anybody to work with, and do not know ow to program, let me know as soon as possible, and we will find an alternative.

(As extra credit problem, write a program for Francis’s algorithm as well, together with an explanation of your code, and apply the algorithm to the same matrix.)

For the remaining problems, you should turn in your own work. You can still collaborate with others, but please make sure to give appropriate credit and indicate clearly who you worked with, what references you consulted, etc:

Give an example of a matrix for which the power method fails. (Include a proof that this is indeed the case.)

Let be an matrix with complex entries. Consider it as a linear transformation . Verify explicitly that is the unique matrix with the property that for all vectors . Recall that for and , their dot product is given by .

Check that if , then the eigenvalues of are real. To see this, let be an eigenvector of , and consider . In particular, this means that the eigenvalues of a symmetric matrix are real.

More significantly, for any Hermitian matrix , that is, one that satisfies , there is a basis consisting of eigenvectors of . We will prove this in due time.

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Thursday, February 28th, 2013 at 3:02 pm and is filed under 403/503: Linear Algebra II. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

The paper MR1029909 (91b:03090). Mekler, Alan H.; Shelah, Saharon. The consistency strength of "every stationary set reflects". Israel J. Math. 67 (1989), no. 3, 353–366, that you mention in the question actually provides the relevant references and explains the key idea of the argument. Note first that $\kappa$ is assumed regular. They refer to MR […]

Start with Conway's base 13 function $c $ (whose range on any interval is all of $\mathbb R $), which is everywhere discontinuous, and note that if $f $ only takes values $0$ and $1$, then $c+f $ is again everywhere discontinuous (since its range on any interval is unbounded). Now note that there are $2^\mathfrak c $ such functions $f $: the characteris […]

Yes, there are such sets. To describe an example, let's start with simpler tasks. If we just want $P\ne\emptyset$ with $P^1=\emptyset$, take $P$ to be a singleton. If we want $P^1\ne\emptyset$ and $P^2=\emptyset$, take $P$ to be a strictly increasing sequence together with its limit $a$. Then $P^1=\{a\}$. If we want $P^2\ne\emptyset$ and $P^3=\emptyset$ […]

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]

No, not even $\mathsf{DC}$ suffices for this. Here, $\mathsf{DC}$ is the axiom of dependent choice, which is strictly stronger than countable choice. For instance, it is a theorem of $\mathsf{ZF}$ that for any set $X$, the set $\mathcal{WO}(X)$ of subsets of $X$ that are well-orderable has size strictly larger than the size of $X$. This is a result of Tarski […]